Plotting $\{z\in \mathbb{C}\mid |z| > \Re(z)-2\}$

Question

How to plot the set of complex numbers
$$\{z\in \mathbb{C}\mid |z| > \Re(z)-2\}$$

I know that $ |z|$ should be a circle centred at $(0,0)$, but I don't know what would be its radius.

Answer 1

Your condition is, with $\;z=x+iy\;,\;\;x,y\in\Bbb R\;$ :

$$\sqrt{x^2+y^2}>x-2\implies x^2+y^2>x^2-4x+4\implies y^2>-4(x-1)$$

You have a parabola there...

Answer 2

This isn't the plot of a complex number: at best, what you wrote is the locus of the points of $\Bbb C$ which satisfy that inequality. Specifically, the locus is the whole complex plane, because of the inequalites $$\lvert z\rvert\ge \lvert \Re z\rvert\ge \Re z>-2+\Re z$$ which hold or all $z\in \Bbb C$.